Subgroups — C₂³×Dic₃₉₈ #27164842

Order 12736.

850 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₉₈	C₂	1	28	yes	maximal, direct product factor
2	C₂³×Dic₁₉₉	C₂	1	2	yes	maximal
3	C₂³×C₄×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₉₈	C₂²	1	112	yes	direct product factor
5	C₂²×Dic₁₉₉	C₂²	1	28	yes	—
6	C₂⁴×C₁₉₉	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₁₉₉	C₂²	1	14	yes	—
8	Dic₃₉₈	C₂³	1	64	yes	direct product factor
9	C₂×Dic₁₉₉	C₂³	1	56	yes	—
10	C₂³×C₁₉₉	Q₈	1	8	yes	complemented
11	C₂³×C₁₉₉	C₂³	1	7	yes	—
12	C₂×C₄×C₁₉₉	C₂³	1	28	yes	—
13	C₂²×C₁₉₉	C₂×Q₈	1	28	yes	complemented
14	C₂²×C₁₉₉	C₂⁴	1	7	yes	—
15	C₄×C₁₉₉	C₂⁴	1	8	yes	—
16	Dic₁₉₉	C₂⁴	1	16	yes	—
17	C₂×C₁₉₉	C₂²×Q₈	1	14	yes	complemented
18	C₂×C₁₉₉	C₂⁵	1	1	yes	characteristic
19	C₁₉₉	C₂³×Q₈	1	1	yes	characteristic, complemented, Sylow
20	C₂³×Q₈	—	199	1	no	maximal, Sylow
21	C₂²×Q₈	—	199	28	no	—
22	C₂³×C₄	—	199	2	no	—
23	C₂³×C₄	D₁₉₉	1	1	yes	characteristic
24	C₂⁴	C₂×D₁₉₉	1	1	yes	characteristic
25	C₂×Q₈	—	199	112	no	—
26	C₂²×C₄	—	199	28	no	—
27	C₂²×C₄	C₂×D₁₉₉	1	14	yes	—
28	C₂³	Dic₃₉₈	1	8	yes	direct product factor
29	C₂³	C₂²×D₁₉₉	1	7	yes	—
30	Q₈	—	199	64	no	—
31	C₂×C₄	—	199	56	no	—
32	C₂×C₄	C₂²×D₁₉₉	1	28	yes	—
33	C₂²	C₂×Dic₃₉₈	1	28	yes	direct product factor
34	C₂²	C₂³×D₁₉₉	1	7	yes	—
35	C₄	—	199	16	no	—
36	C₄	C₂³×D₁₉₉	1	8	yes	—
37	C₂	C₂²×Dic₃₉₈	1	14	yes	direct product factor
38	C₂	C₂⁴×D₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₁₉₉
commutator, cosocle	C₂×C₁₉₉	C₂⁵
Frattini	C₂	C₂⁴×D₁₉₉
Fitting	C₂³×C₄×C₁₉₉	C₂
socle, prime radical	C₂⁴×C₁₉₉	C₂²
hypercenter	C₂³×C₄	D₁₉₉
hypocenter	C₁₉₉	C₂³×Q₈
LCS	G ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×Dic₃₉₈	C₂	direct (N × H)
4. C₂×Dic₃₉₈	C₂²	direct (N × H)
8. Dic₃₉₈	C₂³	direct (N × H)
10. C₂³×C₁₉₉	Q₈	semidirect
13. C₂²×C₁₉₉	C₂×Q₈	semidirect
17. C₂×C₁₉₉	C₂²×Q₈	semidirect
19. C₁₉₉	C₂³×Q₈	semidirect
28. C₂³	Dic₃₉₈	direct (N × H)
33. C₂²	C₂×Dic₃₉₈	direct (N × H)
37. C₂	C₂²×Dic₃₉₈	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×Dic₃₉₈	2
C₂³×Dic₁₉₉	2
C₂³×C₄×C₁₉₉	2
C₂³×Q₈	199

Subgroups — C23×Dic398 #27164842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×Dic₃₉₈ #27164842